Google+ Badge

Wednesday, December 04, 2013

In this paper I introduce the idea of a higher-order modal logic—not a modal logic for higher-order predicate logic, but rather a logic of higher-order modalities. “What is a higher-order modality?”, you might be wondering. Well, if a first-order modality is a way that some entity could have been—whether it is a mereological atom, or a mereological complex, or the universe as a whole—a higher-order modality is a way that a first-order modality could have been. First-order modality is modeled in terms of a space of possible worlds—a set of worlds structured by an accessibility relation, i.e., a relation of relative possibility—each world representing a way that the entire universe could have been. A second-order modality would be modeled in terms of a space of spaces of (first-order) possible worlds, each space representing a way that (first-order) possible worlds could have been. And just as there is a unique actual world which represents the way that things actually are, there is a unique actual space which represents the way that first-order modality actually is.

One might wonder what the accessibility relation itself is like. Presumably, if it is logical or metaphysical modality that is being dealt with, it is reflexive; but is it also symmetric, or transitive? Especially in the case of metaphysical modality, the answer is not clear. And whichever of these properties it may or may not have, could that itself have been different? Could at least some rival modal logics represent different ways that first-order modality could have been?

To be clear, the idea behind my proposal isnot just that some things which are possible or necessary might not have been so at the first order, as determined by the actual accessibility relation, but also that the actual accessibility relation, and hence the nature or structure of actual modality, could have been different at some higher order of modality. Even if the accessibility relation is actually both symmetric and transitive, perhaps it could (second-order) have been otherwise: There is a (second-order) possible space of worlds in which it is different, where it fails to be symmetric, or transitive. We must, therefore, introduce the notion of a higher-order accessibility relation, one that in this case relates spaces of first-order worlds. The question then arises as to whether that relation is symmetric, or transitive. We can then consider third-order modalities, spaces of spaces of spaces of possible worlds, where the second-order accessibility relation differs from how it actually is. I can see no reason why there should be a limit to this hierarchy of higher-order modalities, any more than I can see a reason why there should be a limit to the hierarchy of higher-order properties. There will thus be an infinity of orders, one for each positive integer, and each order will have an accessibility relation of its own. To keep things as clear as possible, a space of first-order points (i.e., of possible worlds) shall be called a galaxy, a space of second-order points, a universe, and a space of any higher order, a cosmos. However, to keep things as simple as possible, in what follows I will deal with but a single cosmos at a time, and hence will not deal with modalities higher than the third order.

The accessibility relation is not the only thing that might be thought to vary between spaces of worlds: Perhaps the contents of the spaces can vary as well. While I presume that the contents of the worlds themselves remain constant—it makes doubtful sense to suppose that in one space some entity e exists in a world w and in another space e doesn’t exist in that same world w—we may suppose that different spaces may differ as to which worlds they contain, just as different worlds may differ as to which objects they contain.Thus we might have a higher-order analogue of a variable-domain modal logic. There seem, then, to be three ways in which spaces can differ: First, as to the properties of the accessibility relation; second, as to which worlds the relation relates; and third, as to which worlds or spaces are parts of their domains.

The paper will be structured as follows. In Section 2 I provide some reasons why one might want to pursue this kind of project in the first place. In Section 3 I outline the syntax and semantics of my proposed logic. Section 4 covers semantic tableaux for this system; and after giving the rules for their construction, I construct a few of them myself to establish some logical consequences of the system and give the reader a feel for how it works. In Section 5 I outline a potential application of my framework to the metalogic of modal logics. In Sections 6, 7 and 8 I explore some of its potential philosophical implications for areas besides logic, namely the philosophy of language; metaphysics, including the metaphysics of modality, the philosophy of time, and laws of nature; and finally the philosophy of religion, before concluding the paper in Section 9.

Sunday, December 01, 2013

Today I had an idea regarding charitable giving that as far as I know hasn't been implemented, at least on a large scale: Why not have "charity cards", credit cards which automatically donate a certain percentage of what you spend to charity whenever you make a purchase? For example, if you spend $100.00 on something and have a 2% threshold, that would automatically generate a corresponding $2 donation to a charity of your choice. I think that would be good because such "microdonations," which most of us wouldn't bother to make separately, would eventually ad up. They would also require no special effort, which I think is a great psychological barrier to giving. If a lot of people ended up using charity cards, it could easily generate millions (or more) in donations. So instead of getting "points" or filer miles, why not have cards that automatically generate microdonations?

It has been noted (by George Lakoff, among others)
that if someone tells you “Don’t think of an elephant!”, it’s pretty hard not
to do it. Though it would take empirical research to fill in the details, the
answer as to why this is so does not seem hard to discern: To understand the
order—to understand what one is not
supposed to think of—one must understand the term ‘elephant’, and thus come to
think of one. I’d wager that in addition to thinking of one an image of an elephant popped into your
head as well. This is probably
because the concept of an elephant is an empirical concept—no crisp, abstract
definition of an elephant comes readily to mind, so a stereotypical image is
needed to make it intelligible. By contrast, if someone were to tell you “Don’t
think of the number 2!” it is less likely that an image would come up, unless
you confuse the numeral ‘2’ with the number 2—or, at least, that the image
would be unlikely to be constant for different people, or for the same person
at different times.

What, though, if someone were to tell you “Don’t
think of a square circle!” Is it so hard to comply in this case?

Well, maybe it is: Do I really know what it is I’m not supposed to think of? If not, I’m
not really complying with the order, because I fail to understand it. I’m
merely doing what it says.
Nevertheless, what interests me here is not our concept of compliance, but
rather that of conceiving or imagining the impossible.

If one can
imagine or conceive of something, what then could one mean by saying that one
can’t understand how it could be the case? Could it be, for instance, that I
can conceive that a square circle exists, but not that the existence of a
square circle is possible? It would
seem not: It is not more impossible
that it is possible that a square
circle exists than that a square circle exists; so if I can conceive the
latter, I should be able to conceive the former as well. If however, I cannot
conceive of the existence of a square circle, I cannot conceive of its
possibility either. That seems fine, but one could ask: If I cannot conceive of
a square circle, nor of its possibility, how could I still conceive of its impossibility? That is, if I truly
cannot conceive of it, how can I say that it is that, and not something else, or nothing at all, of which I am
unable to conceive? If someone tells me that I cannot eat or drink on the
subway, I know what it is that I am being forbidden to do. And if I am informed
that no human being can run as fast as a cheetah, I know what is being declared
to be impossible, and it is relatively easy to form an image of what the
contrary would look like. But if someone tells me that it is impossible for a
circle to be square, or identical to (i.e., the same thing as) a square, how am
I to know what it is that is being
said to not possibly be the case? For if
you were to be told that snorogs are
impossible, you would have every right to ask your alleged informant what a
snorog is. If no answer were forthcoming, the most you could conclude is that
you cannot know whether they are possible or not if you have no idea what they
are supposed to be. The same should hold just as much for “square circles”: If
I simply don’t know what the term means I should conclude, not that they cannot
exist, but rather that I have no idea whether they can or not, if the term is
meaningful at all. And if I do
understand it, it seems—as was said above—that I must also be able to
understand the claim that square circles are possible, as the possibility of
something cannot be any more impossible than the thing itself. If I
nevertheless say that they or their possible existence is inconceivable, I cannot mean that I do not understand the claim that they exist, or that their existence is
possible. What, then, can I mean?

A natural answer would be something like: In virtue of understanding the term
‘square circle’, I “see” or “grasp” that they cannot exist. But what am I seeing here, and how do I see it?

The second question would involve us in the thorny
details of epistemological debates surrounding a priori knowledge, which I will
not enter into. By contrast, it might seem that an answer to the first question
is trivial: I’m seeing that square circles cannot exist! But what we have said
so far should make us suspicious of this answer. To be sure, it is a right answer to our question, if it’s
true that I’m really seeing that, but it’s not the only possible answer, and certainly not the most helpful one. If
someone were to tell you that they can see that snorogs cannot exist, and you
were to ask them what exactly it is to see that, it would not be very
informative to be told that it is simply to see that snorogs cannot exist! If
you don’t know yourself what snorogs are, or what ‘cannot’ means in this
context, such an answer won’t help you one bit. We have two problems: (1) If I
am to “see that something cannot be the case” in virtue of understanding a term, ‘cannot’ should not mean: It is inconceivable that it is the case. What,
then, does it mean? It must be more than mere nomic or physical impossibility,
for one cannot tell something to be nomically or physically impossible merely in virtue of understanding a
certain term. (2) How am I to tell
whether I or someone else truly understands the term? For unless I have
some means of doing so I am at a loss as to how to assess the claim of
impossibility, and can make no progress.

I’ll leave the first question to one side (for now).
As for the second, in all cases it should be possible to specify the meaning of
a term somehow, to give some kind (not necessarily very exact and not
necessarily very informative) of explanation or definition of what it is. In
addition, for empirical terms it should be possible to imagine the
corresponding entity or recognize it in experience.

Can
one specify the meaning of ‘square circle’? It certainly seems so: One can define it as a circle that is also a square, or
as a circle that is identical to (is the same thing as) a square. Alternatively,
if the term ‘spherical cube’ is already understood, (as, e.g., a sphere that is also a cube, or a sphere that is identical to a cube)
one could define a square circle as a
2-dimensional cross-section of a spherical cube that cuts through its center.
Admittedly, these definitions are not very rigorous, but rigorous definitions
cannot be given for most terms in ordinary use, and they are none the worse off
for that.

One can note that as ‘square’ and ‘circle’ (or ‘sphere’
and ‘cube’) are empirical terms, ‘square circle’ should be one too. But whereas it has appeared easy
to define what a square circle would be, it seems much more difficult to
imagine what one would look like. Supposing one were shown a drawing of a disc
with a large square hole in it, or a square solid with a large circular hole in
it, one would naturally reply that that isn’t what one means . But what then
does one mean? It is not as though I have some (vague) image of a square circle
to which they fail to conform, but rather that no image suggests itself. Is it that it would look like something,
only I don’t know what? Or is it that there is no such thing as “what it would
look like”?

Well, if the One True Logic is not paraconsistent (i.e.,
is not non-explosive), then contradictions entail everything. So if square
circles are contradictory—and they seem to be, for squares are square and
circles are not, while square circles are both squares and circles—then every counterfactual beginning “If there
were square circles, they would look like…” is true. If Logic is not
paraconsistent, then, nothing is easier than imaging what a square circle would
look like: Take anything you please, such as a shoe, a ship, some sealing wax,
a cabbage, or a king. If the One True Logic is
paraconsistent, matters are less clear. But suppose some lover of paradoxes
were to come to you, draw a square and a circle next to each other, and ask you
whether a square circle would look like that.

“Like the square or the circle?”, you enquire.

“Like both,”
he replies.

“Well,” you say, “a square circle is supposed to be
a square that’s identical to a circle, but the figures you have drawn don’t
look identical.”

“I grant that the figures I have drawn are not
identical,” he replies, “but that’s the thing about representations: they need
not share the properties of the things they represent. As for identity, I wasn’t
aware that it looked like anything at
all. If it did, one could tell by mere inspection whether properties are
universals or tropes, by ostending different “instances” or “tokens” of the
same type and checking to see—literally see—that
the instances either do or do not have something numerically identical in
common; and one could thus very easily settle that dispute. As things are this
is not possible, so it seems safe to say we cannot perceptually experience relations
of identity. So I think I can safely say that the circle and the square are depicted
as being identical, even though the depicted square and the depicted circle don’t
look identical.

“Identity may not look like anything,” you reply, “but
difference certainly does, and the
square depicted and the circle depicted look different.

“Naturally,” he replies, “for I have drawn a
contradictory object, which is naturally both self identical and self-distinct.
Since I have drawn a square which is identical
to a circle, it differs even from
itself, and it is this difference
which you are picking up on.”

“Well,” you say, more hesitantly, “not only do they
look different, they appear to be in different places. A square circle isn’t
supposed to be a mereological sum of a square in one place and a circle in another,
but one thing in one place which is both square and circular.”

“That,” he replies, “is merely a defect of the
medium. I drew a square shape and a circular shape in different places, but
they are intended to depict exactly the same location. And in any case, just as
the depicted shape is both self-identical and self-distinct, the depicted
location of the shape must also be both self-identical and self-distinct. The
places of the shape(s) are thus both different and not different, and again it
is the difference which you are picking up on.”

“But,” you sputter in exasperation, “I can see that the square and the circle are different from each other, and also that they are not different from themselves,
but what I cannot see is that they
are not different from each other!”

“But of course you can!” he replies, smiling, “for
take the depicted square. You can surely see that it is not different from
itself; that is, that the square is not different from the square. And since
that square is identical to the depicted circle, it follows by Leibniz’s Law
that the depicted circle looks not to be different from the depicted square!”

At this point I suspect you would give up, and let
the lover of paradoxes go on his merry way. But who has won the hypothetical dispute?
Why shouldn’t we say that his drawing of a square circle is a perfectly good
one—that by looking at it we can now tell what one would look like? If we still
insist, as I think most will, that his drawing isn’t a good one, I think the
proper moral may well be a Wittgensteinian one: When we say that we cannot
imagine a square circle, what we’re really doing (whether we realize it or not)
is excluding any purported description of what one would look like from our language-game(s).
We’re saying, in effect, “No matter what anything
looks like, we refuse to call that ‘what a square circle looks like’, and
no matter what anyone draws, we
refuse to call it ‘a good drawing of a square circle.’ And this isn’t to say
that there can’t be some strange looking drawings—one can simply look at a Penrose
triangle or some of M.C. Escher’s works (or for a real life case, one can check
out ‘the waterfall illusion’). One must get away from the idea that behind a
grammatical rule there stands something that cannot be done—unless one only
means that there is a norm that we shouldn’t speak in a certain way. Any image can be described in language in
multiple ways, and what the case of the lover of paradoxes shows, if it shows
anything, is that our current, actual language game forbids certain descriptions
of certain phenomena. Perhaps, as in the case of the waterfall illusion, a
language game which admits of inconsistent modes of description would be more
appropriate (and note that I say more
appropriate, for consistent modes of description are certainly also available).
As with the description ‘the pitch of sweetness’, the (apparent) description ‘the
look / appearance of a square circle’ has been denied a role in our language
game. Just as ‘the pitch of sweetness’ is not something we fail to hear in any ordinary sense, ‘the look / appearance of a
square circle’ is not something that we fail
to see or imagine in any ordinary sense. Why such terms play no role in our language may be hard to say, but
that they do not could be said to be
shown by our reaction to the case of the lover of paradoxes: the simple fact is
that we know how the investigation is to turn out before we begin to inquire. For it is not that the term ‘square
circle’, like the term ‘snorog,’ simply calls up no image for us as a matter of fact—if someone were to
introduce us to a community where things which were regularly called snorogs looked
such-and-such a way, we would accept that easily—it is rather that we know in
advance that we will refuse to apply the term ‘square circle’ to anything.

In this way we avoid the misleading idea of
imagining the impossible as something that we are unable to do—that “our powers
of imagination are unequal to the task” as Wittgenstein put it in the Investigations—as well as avoiding the
equally misleading picture of logical or conceptual necessity whereby we in
effect personify it as the bouncer at the door of Club Reality; of
logical/conceptual necessity as a powerful force which keeps out the riffraff of
impossibilia struggling to get in in order that they may exist. This, at any rate,
is the best I think I can do to give content to Wittgenstein’s idea of logical/conceptual
necessity as being basically linguistic.